Statistical exponential formulas for homogeneous diffusion

Let ${\mathrm{\Delta }}_p^1$ denote the $1$-homogeneous $p$-Laplacian, for $1 \leq p \leq \infty$. This paper proves that the unique bounded, continuous viscosity solution $u$ of the Cauchy problem $$\{\begin{array}{c}{u}_t-(\frac{p}{N+p-2}){\mathrm{\Delta }}_p^{1}u=0forx\in {\mathbb{R}}^N,t>0\\ \\ u(\cdot ,0)=u_0\in BUC({\mathbb{R}}^N)\end{array}$$ is given by the exponential formula $$u(t):=\underset{n\to \mathrm{\infty }}{\lim }{\left(M_p^{t\mathrm{/}n}\right)}^n{u}_0$$ where the statistical operator $M^{h}_{p} \colon BUC( $\mathbb{R}$^{N} ) \to BUC( $\mathbb{R}$^{N} )$ is defined by $$\left(M_p^h\phi \right)(x):=(1-q){\mathrm{median}}_{\mathrm{\partial }B(x,\sqrt{2h})}\left\{\phi \right\}+q{\mathrm{mean}}_{\mathrm{\partial }B(x,\sqrt{2h})}\left\{\phi \right\}$$ with $q:=\frac{N(p-1)}{N+p-2}$, when $1 \leq p \leq 2$ and by $$\left(M_p^h\phi \right)(x):=(1-q){\mathrm{midrange}}_{\mathrm{\partial }B(x,\sqrt{2h})}\left\{\phi \right\}+q{\mathrm{mean}}_{\mathrm{\partial }B(x,\sqrt{2h})}\left\{\phi \right\}$$ with $q=\frac{N}{N+p-2}$, when $p \geq 2$. Possible extensions to problems with Dirichlet boundary conditions and to homogeneous diffusion on metric measure spaces are mentioned briefly.